Ltl to Buechi Automata

7/24/2019 Ltl to Buechi Automata

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LTL to Bchi Automata

LTL to Bchi AutomataFlavio Lerda


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LTL Formulas

Subset of CTL* Distinct from CTL

AFG LTL

f CTL ! f " AFG Contains a sin#le universal $uantifier

The ath formula f holds for ever% ath

Commonl%& A is omitted G is relaced b% 'bo( or al)a%s F is relaced b% 'diamond or eventuall%


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+(amles of LTL formulas

Al)a%s eventuall% &

AGF or AG AF

Al)a%s after eventuall% $ ' $

AG ' ,- F $ or AG ' ,- AF $

Fairness ' A ''GF .ot a CTL formula


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LTL Semantics

Derived from the CTL* semantics Given an infinite e(ecution trace/s0s12

3 's03 4 4' 3 3 1 53 13 53 153 13 53 i0 i3 3 i0 i3 3 16 5 i0 i3 5 0 78 i 73 1

i

is the suffi( of startin# at si


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Bchi Automata

Automaton )hich accets infinite traces A Bchi automatonis 9,tuleS: ;:: F

S is a finite set of states ;S is a set of initial states SS is a transition relation

FS is a set of accetin# states

An infinite se$uence of statesis accetediff it contains accetin# states infinitel%often


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+(amle

S0 S1 S5

1/S0S1S5S5S5S52

5/S0S1S5S1S5S12


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LTL and Bchi Automata

LTL formula =eresents a set of infinite traces)hich

satisf% such formula

Bchi AutomatonAccets a set of infinite traces

>e can build an automaton )hich accetsall and onl%the infinite traces reresentedb% an LTL formula


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Labeled Bchi Automata

Given a set of atomic roosition? Define a labelin# function

& S 5?

+ach state is assi#ned a set of roositionsthat must be true

Similar to the labelin# for the model @


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Generatin# Bchi Automata

>e need a rocedureto #enerate a Bchiautomaton #iven an LTL formula +fficientl%

Formulas are usuall% small Bchi automaton exponential in the size of the formula

The cost of model checin# is proportionalto the sie of theautomaton

.on,deterministic Bchi automata are not equivalent

to deterministic Bchi automata Cannotuse automata minimiation al#orithms Findin# the minimal automata is e(onential


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Aroach

Formula rewriting =e)rite the formula in negated normal formAl% re)ritin# rules

Core translation Turns an LTL formula into a generalized Bchi

automaton

Degeneralization Turns a #eneralied Bchi automaton into aBchi automaton


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=e)ritin#

Negate normal form .e#ation aears onlyin front of literals 6se the follo)in# identities

44/ 4G / F 4 4F / G 4 4'6 / '4 '4

4' / '4 6 '4 'sometimes = is the Release oerator

Dual of Until

ALTL BhiA


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=e)ritin#

Additional rewriting rules The% are not guaranteedto %ield smaller

automata

The sie of the automaton is e(onential inthe sie of the formula

+(amles

' 6 ' '6 ' ' ' GF GF GF '

15820ALTL BhiA


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=e)ritin#

The core al#orithm onlyhandles ! ! "! #

6se the follo)in#& F T6 G 4F 44'T6 4 / F

15820ALTLtBhiA t t


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Core Translation

$ea

@ae use of the follo)in#6 ''6 ''

15820ALTLtBhiA t t


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+(amleF

p

(TUp)

Old:{}

New:{TUp}

Next:{}

Old:{TUp}New:{T}

Next:{TUp}

Old:{TUp}New:{p}

Next:{}

Old:{TUp}New:{}

Next:{TUp}

Old:{TUp}Next:{TUp}

Old:{TUp, p}New:{}

Next:{}

Old:{TUp, p}Next:{}

T pp

Old:{}New:{}

Next:{}

Old:{}Next:{}

15820ALTLtBhiA t t


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Core Translation

Noe =eresent a sub,formula Contain information about the past, the

present and the future

%tate =eresents a state in the final automaton The% are the nodes that have full& expane

15820ALTLtoBchiAutomata


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Core Translation

+(ansion Select a formula from the New field ;f it is a literal: add it to the Oldfield

Ether)ise

'.e):.e(tand '.e):.e(t6

'.e):.e)6 and '.e):.e(t

'.e):.e) and '.e)::.e(t

U (X(U ))

V (X(V ))



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Core Translation

Noes to states ;f a node has no New formulas Create a ne) node )ith all the Nextformulas Create an ed#e bet)een the t)o nodes Chec if there is an% e$uivalent state

>ith the same Next field

>ith the same Old field



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Core Translation

A''epting states Generalied Bchi automaton

Multipleaccetin# sets Ene for each Until sub,formula '6 Such that

The Oldfield doesnHt contain 6

or The Oldfield does contain



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De#eneraliation

Turn a #eneralied Bchi automaton into a()'hi automaton

Consider as man& 'opiesof the automaton as

the number of accepting sets =elace incomin# ed#es from accetin# states

)ith ed#es to the next co%

+ach c%cle must #o throu#h everyco% +ach c%cle must contains accetin# states fromeach accepting set



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+(amle

T

a b

T

a b

T

1

1,2

2

F a F b



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+(amle

T

a b

T

a

T

T

a b

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b

T

15-820ALTLtoBchiAutomata


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+(amle

T

a b

T

a

T

T

a b

T

b

T



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+(amle

T

a b

T

a

T

T

a b

T



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+(amle

T

a b

T

a

T T



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+(amle

T

a b

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a

TT



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Etimiations

Can be done at ea'h stage Tr% to minimie

The number of states and transitions

The number of accepting states ;nvolve Stron#l% connected comonents Fair 'bisimulation

Expensivebut The Bchi automaton is usuall% small The savin# durin# verification can be ver% hi#h

Ltl to Buechi Automata

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